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Let us now apply what we have just learned about the mean, variance, and
standard deviation of a general distribution function
to the specific case of the
binomial distribution function. Recall,
that if a simple system has just two possible outcomes,
denoted 1 and 2, with
respective probabilities and ,
then the probability of obtaining
occurrences of outcome 1 in observations is

(38) 
Thus, the mean number of occurrences of outcome 1 in observations
is given by

(39) 
This is a rather nasty looking expression! However, we can see that if the
final factor
were absent, it would just reduce to the binomial expansion, which we
know how to sum. We can take advantage of this fact by using a rather elegant
mathematical sleight of hand. Observe that since

(40) 
the summation can be rewritten as

(41) 
This is just algebra, and has nothing to do with probability theory.
The term in square brackets is the familiar binomial expansion, and
can be written more succinctly as .
Thus,

(42) 
However, for the case in hand, so

(43) 
In fact, we could have guessed this result.
By definition, the probability is the number of occurrences of the
outcome 1 divided by the number of trials, in the limit as the number
of trials goes to infinity:

(44) 
If we think carefully, however,
we can see that taking the limit as the number
of trials goes to infinity is equivalent to taking the mean value,
so that

(45) 
But, this is just a simple rearrangement of Eq. (43).
Let us now calculate the variance of . Recall that

(46) 
We already know
,
so we just need to calculate
.
This average is written

(47) 
The sum can be evaluated using a simple extension of the mathematical trick
we used earlier to evaluate
. Since

(48) 
then
Using yields
since
. It follows that the variance
of is given by

(51) 
The standard deviation of is just the square root of the variance, so

(52) 
Recall that this quantity is essentially the width of the range over which
is distributed around its mean value. The relative width of the
distribution is characterized by

(53) 
It is clear from this formula that the relative width decreases like
with increasing . So, the greater the number of trials, the
more likely it is that an observation of will yield a result
which is relatively close to the mean value
. This
is a very important result.
Next: The Gaussian distribution
Up: Probability theory
Previous: The mean, variance, and
Richard Fitzpatrick
20060202